magnetohydrodynamics in stationary and axisymmetric...

Magnetohydrodynamicsin stationary and axisymmetric spacetimes:

a geometrical approach

Eric Gourgoulhon1 Charalampos Markakis2, Koji Uryu3 & Yoshiharu Eriguchi4

1LUTH, Observatoire de Paris / CNRS / Univ. Paris Diderot, Meudon, France

2Dept. of Physics, University of Wisconsin, Milwaukee, USA

3Dept. of Physics, University of the Ryukyus, Okinawa, Japan

4Dept. of Earth Science and Astronomy, University of Tokyo, Japan

Seminar at GReCO, IAP, Paris7 March 2011

E. Gourgoulhon, C. Markakis, K. Uryu & Y. Eriguchi () Stationary & axisymmetric GRMHD GReCO, IAP, Paris, 7 March 2011 1 / 47

Plan

1 Introduction

2 Relativistic MHD with exterior calculus

3 Stationary and axisymmetric electromagnetic fields in general relativity

4 Stationary and axisymmetric MHD

5 Some subcases of the master transfield equation

6 Conclusion


Introduction

Outline

1 Introduction





6 Conclusion


Introduction

Short history of general relativistic MHDfocusing on stationary and axisymmetric spacetimes

Lichnerowicz (1967): formulation of GRMHD

Bekenstein & Oron (1978), Carter (1979) : development of GRMHD forstationary and axisymmetric spacetimes

Mobarry & Lovelace (1986) : Grad-Shafranov equation for Schwarzschildspacetime

Nitta, Takahashi & Tomimatsu (1991), Beskin & Pariev (1993) :Grad-Shafranov equation for Kerr spacetime

Ioka & Sasaki (2003) : Grad-Shafranov equation in the most general (i.e.noncircular) stationary and axisymmetric spacetimes

NB: not speaking about numerical GRMHD here(see e.g. Shibata & Sekiguchi (2005))


Introduction

Why a geometrical approach ?

Previous studies made use of component expressions, the covariance of whichis not obviousFor instance, two of main quantities introduced by Bekenstein & Oron (1978)and employed by subsequent authors are

ω := −F01

F31and C :=

F31√−gnu2

GRMHD calculations can be cumbersome by means of standard tensorcalculus

On the other side

As well known, the electromagnetic field tensor F is fundamentally a 2-formand Maxwell equations are most naturally expressible in terms of the exteriorderivative operator

The equations of perfect hydrodynamics can also be recast in terms ofexterior calculus, by introducing the fluid vorticity 2-form (Synge 1937,Lichnerowicz 1941)

Cartan’s exterior calculus makes calculations easier !


Introduction

Exterior calculus in one slide

A p-form (p = 0, 1, 2, . . .) is a multilinear form (i.e. a tensor 0-timescontravariant and p-times covariant: ωα1...αp

) that is fully antisymmetric

Index-free notation: given a vector ~v and a p-form ω, ~v · ω and ω · ~v arethe (p− 1)-forms defined by

~v · ω := ω(~v, ., . . . , .) [ (~v · ω)α1···αp−1 = vµωµα1···αp−1 ]ω · ~v := ω(., . . . , ., ~v) [(ω · ~v)α1···αp−1 = ωα1···αp−1µv

µ ]

Exterior derivative : p-form ω 7−→ (p+ 1)-form dω such that

0-form : (dω)α = ∂αω

1-form : (dω)αβ = ∂αωβ − ∂βωα2-form : (dω)αβγ = ∂αωβγ + ∂βωγα + ∂γωαβ

The exterior derivative is nilpotent: ddω = 0A very powerful tool : Cartan’s identity expressing the Lie derivative of a

p-form along a vector field: L~v ω = ~v · dω + d(~v · ω)


Relativistic MHD with exterior calculus

Outline

1 Introduction





6 Conclusion



General framework and notations

Spacetime:

M : four-dimensional orientable real manifold

g : Lorentzian metric on M , sign g = (−,+,+,+)ε : Levi-Civita tensor (volume element 4-form) associated with g:for any orthonormal basis (~eα),

ε(~e0, ~e1, ~e2, ~e3) = ±1

ε gives rise to Hodge duality : p-form 7−→ (4− p)-form

Notations:

~v vector =⇒ v 1-form associated to ~v by the metric tensor:

v := g(~v, .) [v = v[] [uα = gαµuµ]

ω 1-form =⇒ ~ω vector associated to ω by the metric tensor:

ω =: g(~ω, .) [~ω = ω]] [ωα = gαµωµ]



Maxwell equations

Electromagnetic field in M : 2-form F which obeys to Maxwell equations:

dF = 0

d ?F = µ0 ?j

dF : exterior derivative of F : (dF )αβγ = ∂αFβγ + ∂βFγα + ∂γFαβ

?F : Hodge dual of F : ?Fαβ :=12εαβµνF

µν

?j : 3-form Hodge-dual of the 1-form j associated to the electric 4-current~j : ?j := ε(~j, ., ., .)µ0 : magnetic permeability of vacuum



Electric and magnetic fields in the fluid frame

Fluid : congruence of worldlines in M =⇒ 4-velocity ~u

Electric field in the fluid frame: 1-form e = F · ~u

Magnetic field in the fluid frame: vector ~b such that b = ~u · ?F

e and ~b are orthogonal to ~u : e · ~u = 0 and b · ~u = 0

F = u ∧ e+ ε(~u,~b, ., .)

?F = −u ∧ b+ ε(~u, ~e, ., .)



Perfect conductor

Fluid is a perfect conductor ⇐⇒ ~e = 0 ⇐⇒ F · ~u = 0From now on, we assume that the fluid is a perfect conductor (ideal MHD)

The electromagnetic field is then entirely expressible in terms of vectors ~u and ~b:

F = ε(~u,~b, ., .)

?F = b ∧ u



Alfven’s theorem

Cartan’s identity applied to the 2-form F :

L~u F = ~u · dF + d(~u · F )

Now dF = 0 (Maxwell eq.) and ~u · F = 0 (perfect conductor)Hence the electromagnetic field is preserved by the flow:

L~u F = 0

Application:d

dτ

∮C(τ)

A = 0

τ : fluid proper time

C(τ) = closed contour dragged along by the fluid

A : electromagnetic 4-potential : F = dA

Proof:d

dτ

∮C(τ)

A =d

dτ

∫S(τ)

dA︸︷︷︸F

=d

dτ

∫S(τ)

F =∫S(τ)

L~u F︸︷︷︸0

= 0

Non-relativistic limit:

∫S~b · d~S = const ← Alfven’s theorem (mag. flux freezing)



Perfect fluid

From now on, we assume that the fluid is a perfect one: its energy-momentumtensor is

T fluid = (ε+ p)u⊗ u+ pg

Simple fluid model: all thermodynamical quantities depend on

s: entropy density in the fluid frame,

n: baryon number density in the fluid frame

Equation of state : ε = ε(s, n) =⇒

T :=

∂ε

∂stemperature

µ :=∂ε

∂nbaryon chemical potential

First law of thermodynamics =⇒ p = −ε+ Ts+ µn

=⇒ enthalpy per baryon : h =ε+ p

n= µ+ TS , with S :=

s

n(entropy per

baryon)



Conservation of energy-momentum

Conservation law for the total energy-momentum:

∇· (T fluid + T em) = 0 (1)

From Maxwell equations, ∇· T em = −F · ~jUsing baryon number conservation, ∇· T fluid can be decomposed in twoparts:

along ~u: ~u ·∇· T fluid = −nT ~u · dSorthogonal to ~u : ⊥u∇· T fluid = n[~u · d(hu)− TdS]

[Synge 1937] [Lichnerowicz 1941] [Taub 1959] [Carter 1979]

Ω := d(hu) vorticity 2-form

Since ~u · F · ~j = 0, Eq. (1) is equivalent to the system

~u · dS = 0 (2)

~u · d(hu)− TdS =1nF · ~j (3)

Eq. (3) is the MHD-Euler equation in canonical formE. Gourgoulhon, C. Markakis, K. Uryu & Y. Eriguchi () Stationary & axisymmetric GRMHD GReCO, IAP, Paris, 7 March 2011 14 / 47

Stationary and axisymmetric electromagnetic fields in general relativity

Outline

1 Introduction





6 Conclusion



Stationary and axisymmetric spacetimes

Assume that (M , g) is endowed with two symmetries:1 stationarity : ∃ a group action of (R,+) on M such that

the orbits are timelike curvesg is invariant under the (R, +) action :

if ~ξ is a generator of the group action,

L~ξ g = 0 (4)

2 axisymmetry : ∃ a group action of SO(2) on M such thatthe set of fixed points is a 2-dimensional submanifold ∆ ⊂M (called therotation axis)g is invariant under the SO(2) action :if ~χ is a generator of the group action,

L~χ g = 0 (5)

(4) and (5) are equivalent to Killing equations:

∇αξβ +∇βξα = 0 and ∇αχβ +∇βχα = 0



Stationary and axisymmetric spacetimes

No generality is lost by considering that the stationary and axisymmetric actionscommute [Carter 1970] :(M , g) is invariant under the action of the Abelian group (R,+)× SO(2), andnot only under the actions of (R,+) and SO(2) separately. It is equivalent to saythat the Killing vectors commute:

[~ξ, ~χ] = 0

=⇒ ∃ coordinates (xα) = (t, x1, x2, ϕ) on M such that ~ξ =∂

∂tand ~χ =

∂

∂ϕWithin them, gαβ = gαβ(x1, x2)

Adapted coordinates are not unique:

t′ = t+ F0(x1, x2)x′

1 = F1(x1, x2)x′

2 = F2(x1, x2)ϕ′ = ϕ+ F3(x1, x2)



Stationary and axisymmetric electromagnetic field

Assume that the electromagnetic field is both stationary and axisymmetric:

L~ξ F = 0 and L~χ F = 0 (6)

Cartan’s identity and Maxwell eq. =⇒ L~ξ F = ~ξ · dF︸︷︷︸0

+d(~ξ · F ) = d(~ξ · F )

Hence (6) is equivalent to

d(~ξ · F ) = 0 and d(~χ · F ) = 0

Poincare lemma =⇒ ∃ locally two scalar fields Φ and Ψ such that

~ξ · F = −dΦ and ~χ · F = −dΨ

Link with the 4-potential A: one may use the gauge freedom on A to set

Φ = A · ~ξ = At and Ψ = A · ~χ = Aϕ



Symmetries of the scalar potentials

From the definitions of Φ and Ψ:

L~ξ Φ = ~ξ · dΦ = −F (~ξ, ~ξ) = 0

L~χΨ = ~χ · dΨ = −F (~χ, ~χ) = 0

L~χ Φ = ~χ · dΦ = −F (~ξ, ~χ)

L~ξ Ψ = ~ξ · dΨ = −F (~χ, ~ξ) = F (~ξ, ~χ)

We have d[F (~ξ, ~χ)] = d[~ξ · dΨ] = L~ξ dΨ− ~ξ · ddΨ︸︷︷︸0

= L~ξ (F · ~χ) = 0

Hence F (~ξ, ~χ) = const

Assuming that F vanishes somewhere in M (for instance at spatial infinity), weconclude that

F (~ξ, ~χ) = 0

Then L~ξ Φ = L~χΦ = 0 and L~ξ Ψ = L~χΨ = 0

i.e. the scalar potentials Φ and Ψ obey to the two spacetime symmetries



Most general stationary-axisymmetric electromagnetic field

F = dΦ ∧ ξ∗ + dΨ ∧ χ∗ +I

σε(~ξ, ~χ, ., .) (7)

?F = ε( ~∇Φ, ~ξ∗, ., .) + ε( ~∇Ψ, ~χ∗, ., .)− I

σξ ∧ χ (8)

with

ξ∗ :=1σ

(−X ξ +Wχ

), χ∗ :=

1σ

(W ξ + V χ

)V := −ξ · ~ξ , W := ξ · ~χ , X := χ · ~χ , σ := V X +W 2

[Carter (1973) notations]

I := ?F (~ξ, ~χ) ← the only non-trivial scalar, apart from F (~ξ, ~χ), one can

form from F , ~ξ and ~χ

(ξ∗,χ∗) is the dual basis of (~ξ, ~χ) in the 2-plane Π := Vect(~ξ, ~χ) :

ξ∗ · ~ξ = 1, ξ∗ · ~χ = 0, χ∗ · ~ξ = 0, χ∗ · ~χ = 1∀~v ∈ Π⊥, ξ∗ · ~v = 0 and χ∗ · ~v = 0



Most general stationary-axisymmetric electromagnetic fieldThe proof

Consider the 2-form H := F − dΦ ∧ ξ∗ − dΨ ∧ χ∗It satisfies

H(~ξ, .) = F (~ξ, .)︸︷︷︸−dΦ

−(~ξ · dΦ︸︷︷︸0

)ξ∗ + (ξ∗ · ~ξ︸︷︷︸1

)dΦ− (~ξ · dΨ︸︷︷︸0

)χ∗ + (χ∗ · ~ξ︸︷︷︸0

)dΨ = 0

Similarly H(~χ, .) = 0. Hence H|Π = 0

On Π⊥, H|Π⊥ is a 2-form. Another 2-form on Π⊥ is ε(~ξ, ~χ, ., .)∣∣∣Π⊥

Since dim Π⊥ = 2 and ε(~ξ, ~χ, ., .)∣∣∣Π⊥6= 0, ∃ a scalar field I such that

H|Π⊥ =I

σε(~ξ, ~χ, ., .)

∣∣∣Π⊥

. Because both H and ε(~ξ, ~χ, ., .) vanish on Π, we

can extend the equality to all space:

H =I

σε(~ξ, ~χ, ., .)

Thus F has the form (7). Taking the Hodge dual gives the form (8) for ?F , on

which we readily check that I = ?F (~ξ, ~χ), thereby completing the proof.E. Gourgoulhon, C. Markakis, K. Uryu & Y. Eriguchi () Stationary & axisymmetric GRMHD GReCO, IAP, Paris, 7 March 2011 21 / 47


Example: Kerr-Newman electromagnetic field

Using Boyer-Lindquist coordinates (t, r, θ, ϕ), the electromagnetic field of theKerr-Newman solution (charged rotating black hole) is

F =µ0Q

4π(r2 + a2 cos2 θ)2

[(r2 − a2 cos2 θ) dr − a2r sin 2θ dθ

]∧ dt

+[a(a2 cos2 θ − r2) sin2 θ dr + ar(r2 + a2) sin 2θ dθ

]∧ dϕ

Q: total electric charge, a := J/M : reduced angular momentum

For Kerr-Newman, ξ∗ = dt and χ∗ = dϕ; comparison with (7) leads to

Φ = −µ0Q

4πr

r2 + a2 cos2 θ, Ψ =

µ0Q

4πar sin2 θ

r2 + a2 cos2 θ, I = 0

Non-rotating limit (a = 0): Reissner-Nordstrom solution: Φ = −µ0

4πQ

r, Ψ = 0



Maxwell equations

First Maxwell equation: dF = 0

It is automatically satisfied by the form (7) of F

Second Maxwell equation: d ?F = µ0 ?j

It gives the electric 4-current:

µ0~j = a ~ξ + b ~χ− 1

σ~ε(~ξ, ~χ, ~∇I, .) (9)

with

a := ∇µ(X

σ∇µΦ− W

σ∇µΨ

)+

I

σ2[−XCξ +WCχ]

b := −∇µ(W

σ∇µΦ +

V

σ∇µΨ

)+

I

σ2[WCξ + V Cχ]

Cξ := ?(ξ ∧ χ ∧ dξ) = εµνρσξµχν∇ρξσ (1st twist scalar)

Cχ := ?(ξ ∧ χ ∧ dχ) = εµνρσξµχν∇ρχσ (2nd twist scalar)

Remark: ~j has no meridional component (i.e. ~j ∈ Π) ⇐⇒ dI = 0E. Gourgoulhon, C. Markakis, K. Uryu & Y. Eriguchi () Stationary & axisymmetric GRMHD GReCO, IAP, Paris, 7 March 2011 23 / 47


Simplification for circular spacetimes

Spacetime (M , g) is circular ⇐⇒ the planes Π⊥ are integrable in 2-surfaces⇐⇒ Cξ = Cχ = 0

Generalized Papapetrou theorem [Papapetrou 1966] [Kundt & Trumper 1966] [Carter 1969] :a stationary and axisymmetric spacetime ruled by the Einstein equation is circulariff the total energy-momentum tensor T obeys to

ξµT [αµ ξβχγ] = 0

χµT [αµ ξβχγ] = 0

Examples:

circular spacetimes: Kerr-Newman, rotating star, magnetized rotating starwith either purely poloidal magnetic field or purely toroidal magnetic field

non-circular spacetimes: rotating star with meridional flow, magnetizedrotating star with mixed magnetic field

In what follows, we do not assume that (M , g) is circular


Stationary and axisymmetric MHD

Outline

1 Introduction





6 Conclusion



Perfect conductor hypothesis (1/2)

F · ~u = 0with the fluid 4-velocity decomposed as

~u = λ(~ξ + Ω~χ) + ~w, ~w ∈ Π⊥ (10)

Ω is the rotational angular velocity and ~w is the meridional velocity

u · ~u = −1 ⇐⇒ λ =

√1 +w · ~w

V − 2ΩW − Ω2X

We haveL~u Φ = 0 and L~uΨ = 0 , (11)

i.e. the scalar potentials Φ and Ψ are constant along the fluid lines.

Proof: L~uΦ = ~u · dΦ = −F (~ξ, ~u) = 0 by the perfect conductor property.

Corollary: since we had already L~ξ Φ = L~χΦ = 0 and L~ξ Ψ = L~χΨ = 0, it

follows from (11) that

~w · dΦ = 0 and ~w · dΨ = 0 (12)



Perfect conductor hypothesis (2/2)

Expressing the condition F · ~u = 0 with the general form (7) of astationary-axisymmetric electromagnetic field yields

(ξ∗ · ~u︸︷︷︸λ

)dΦ− (dΦ · ~u︸︷︷︸0

)ξ∗ + (χ∗ · ~u︸︷︷︸λΩ

)dΨ− (dΨ · ~u︸︷︷︸0

)χ∗ +I

σε(~ξ, ~χ, ., ~u)︸︷︷︸−ε(~ξ,~χ, ~w,.)

= 0

Hence

dΦ = −Ω dΨ +I

σλε(~ξ, ~χ, ~w, .) (13)



Conservation of baryon number and stream function

Baryon number conservation : ∇ · (n~u) = 0 ⇐⇒ d(n ?w) = 0

→Poincare Lemma: ∃ a 2-form H such that n ?w = dH

Considering the scalar field f := H(~ξ, ~χ), we get

df = n ε(~ξ, ~χ, ~w, .) ⇐⇒ ~w = − 1σn

~ε(~ξ, ~χ, ~∇f, .) (14)

f is called the (Stokes) stream function

It follows from (14) that

~ξ · df = 0 and ~χ · df = 0 =⇒ f obeys to the spacetime symmetries

~u · df = 0 =⇒ f is constant along any fluid line

The perfect conductivity relation (13) is writable as

dΦ = −Ω dΨ +I

σnλdf (15)



Integrating the MHD-Euler equation

With the writing (10) of ~u, (7) of F and (9) of ~j, the MHD-Euler equation

~u · d(hu)− TdS =1nF · ~j

can be shown to be equivalent to the system

~w · d(hu · ~ξ)− 1µ0σn

ε(~ξ, ~χ, ~∇I, ~∇Φ) = 0 (16)

~w · d(hu · ~χ)− 1µ0σn

ε(~ξ, ~χ, ~∇I, ~∇Ψ) = 0 (17)

λd(hu · ~ξ) + λΩ d(hu · ~χ)− 1n

[q +

λh

σ(Cξ + ΩCχ)

]df − I

µ0σndI

+ξ∗ · ~jn

dΦ +χ∗ · ~jn

dΨ + T dS = 0. (18)

with q := −∇µ(h

σ n∇µf

)E. Gourgoulhon, C. Markakis, K. Uryu & Y. Eriguchi () Stationary & axisymmetric GRMHD GReCO, IAP, Paris, 7 March 2011 29 / 47


Introducing the master potential (1/2)

As a consequence of the perfect conductivity properties (12) and the baryonnumber conservation relation (14), one has

ε(~ξ, ~χ, ~∇f, ~∇Φ) = 0 and ε(~ξ, ~χ, ~∇f, ~∇Ψ) = 0Along with Eq. (15) above, i.e.

dΦ = −Ω dΨ +I

σnλdf

this implies thatThe gradient 1-forms dΦ, dΨ and df are colinear to each other

Standard approach in GRMHD: privilege Ψ and write dΦ = −ω dΨ, df = adΨDrawback: This is degenerate if dΨ = 0

Here: we follow the approach of Tkalich (1959) and Soloviev (1967) forNewtonian MHD, i.e. we introduce a fourth potential Υ such that

1 Υ obeys to both spacetime symmetries2 dΥ 6= 03 ∃ three scalar fields α, β and γ such that

dΦ = αdΥ, dΨ = β dΥ, df = γ dΥ






dΦ = −Ω dΨ +I

σnλdf


Standard approach in GRMHD: privilege Ψ and write dΦ = −ω dΨ, df = adΨ

Drawback: This is degenerate if dΨ = 0









dΦ = −Ω dΨ +I

σnλdf


Standard approach in GRMHD: privilege Ψ and write dΦ = −ω dΨ, df = adΨDrawback: This is degenerate if dΨ = 0








All potentials can be considered as functions of Υ:

Φ = Φ(Υ), Ψ = Ψ(Υ), f = f(Υ),α = Φ′(Υ), β = Ψ′(Υ), γ = f ′(Υ)

Proof: ddΦ = 0 = dα ∧ dΥ =⇒ α = α(Υ) =⇒ Φ = Φ(Υ) with α = Φ′

Υ is conserved along the fluid lines (since f is)

the perfect conductor property (13) leads to the relation

α+ Ωβ =γI

σnλ



Integrating the first two equations of the MHD-Eulersystem

Expressing ~w in terms of df via (14) and using df = γ dΥ as well as dΦ = αdΥenables us to write the first equation of the MHD-Euler system [Eq. (16)] in theequivalent form

ε

(~ξ, ~χ, ~∇Υ, −γ ~∇(hu · ~ξ) +

α

µ0

~∇I

)= 0

=⇒ −γhu · ~ξ + αI/µ0 must be a function of Υ, Σ(Υ) say:

Σ(Υ) = −γhu · ~ξ +αI

µ0= γλh(V −WΩ) +

αI

µ0

Similarly, the second equation of the MHD-Euler system [Eq. (17)] is equivalent tothe existence of a function Λ(Υ) such that

Λ(Υ) = γhu · ~χ− βI

µ0= γλh(W +XΩ)− βI

µ0



Interpretation as Bernoulli-like theorems

If the fluid motion is purely rotational, ~u = λ(~ξ + Ω~χ) and any scalarquantity obeying to the two spacetime symmetries is conserved along thefluid lines

If the fluid flow has some meridional component, ~w 6= 0 ⇐⇒ df 6= 0: we

may choose Υ = f ; then γ = 1 and

Σ = λh(V −WΩ) +αI

µ0Λ = λh(W +XΩ)− βI

µ0

We recover the two streamline-constants of motion found by Bekenstein &Oron (1978) in a slightly more complicated form:

Σ = −(h+

|b|2

µ0n

)u · ~ξ − β

µ0

[u ·(~ξ − α

β~χ

)](b · ~ξ)

Λ =(h+

|b|2

µ0n

)u · ~χ+

β

µ0

[u ·(~ξ − α

β~χ

)](b · ~χ)



Non-relativistic limit

At the Newtonian limit and in standard isotropic spherical coordinates (t, r, θ, ϕ),V = 1 + 2Φgrav, W = 0X = (1− 2Φgrav)r2 sin2 θσ = r2 sin2 θ,

where Φgrav is the Newtonian gravitational potential (|Φgrav| 1)

Moreover, introducing the mass density ρ := mb n (mb mean baryon mass)

and specific enthalpy H :=εint + p

ρ, we get h = mb(1 +H) with H 1

Then

Σmb− 1 = H + Φgrav +

v2

2+

αI

µ0mb(when I = 0, classical Bernoulli theorem)

Λmb

= Ω r2 sin2 θ − βI

µ0mb



Entropy as a function of the master potential

Equation (2) (resulting from ∇ · T = 0) implies successively

~u · dS = 0 =⇒ ~w · dS = 0 =⇒ ε(~ξ, ~χ, ~∇f, ~∇S) = 0

If df 6= 0, this implies S = S(f), i.e.

S = S(Υ) (19)

If df = 0 (purely rotational flow), we assume that (19) still holds



The master transfield equation

Thanks to the existence of Σ(Υ), Λ(Υ) and S(Υ), the remaining part of the

MHD-Euler equation [Eq. (18)] can be rewritten as A dΥ = 0 . Since dΥ 6= 0,it is equivalent to A = 0. Expressing A , we get the master transfield equation:

A∆∗Υ + nh

[γ2d

(hn

)− 1

µ0

(β2dV + 2αβdW − α2dX

)]· ~∇Υ

+γγ′ − n

µ0h[V ββ′ +W (α′β + αβ′)−Xαα′]

dΥ · ~∇Υ

+σn2

h

λγ

[ΩΛ′ − Σ′ + I

µ0(α′ + Ωβ′) + γ′λh(V − 2WΩ−XΩ2)

]+ TS′

−γλn (Cξ + ΩCχ) + In

µ0σh[(Wβ −Xα)Cξ + (Wα+ V β)Cχ] = 0

(20)

with A := γ2 − n

µ0h(V β2 + 2Wαβ −Xα2) and ∆∗Υ := σ∇µ

(1σ∇µΥ

)Eq. (20) is called transfield for it expresses the component along dΥ of theMHD-Euler equation and dΥ is transverse to the magnetic field in the fluid frame~b, in the sense that ~b · dΥ = 0



Poloidal wind equation

The master transfield eq. has to be supplemented by the poloidal wind equation,arising from the 4-velocity normalization u · ~u = −1, with λ and Ω expressed interms of α, β, γ, Σ, Λ and h :

h2

(σ +

γ2

n2dΥ · ~∇Υ

)− 1γ2

(XΣ2 + 2WΣΛ− V Λ2

)+

n

µ0h

A+ γ2

A2γ2[(Xα−Wβ)Σ + (V β +Wα)Λ]2 = 0

(21)

Notice that I, λ and Ω in Eq. (20) can be expressed in terms of α, β, γ, Σ, Λ, nand h. Then

Given

the metric (represented by V , X, W , σ and ∇),

the EOS h = h(n, S),

the six functions α(Υ), β(Υ), γ(Υ), Σ(Υ), Λ(Υ) and S(Υ),

Eqs. (20)-(21) constitute a system of 2 PDEs for the 2 unknowns (Υ, n)

Solving it provides a complete solution of the MHD-Euler equation


Some subcases of the master transfield equation

Outline

1 Introduction





6 Conclusion



Subcase 1 : Newtonian limit

Expression of Σ and Λ:

Σ = γmb

(1 +H + Φgrav +

v2

2

)+αI

µ0

Λ = γmbr2 sin2 θΩ− βI

µ0

Master transfield equation:

A∆∗Υ− γ2

ndn · ~∇Υ +

(γγ′ − n

µ0mbββ′)

dΥ · ~∇Υ

+r2 sin2 θn2

mb

1γ

[ΩΛ′ − Σ′ +

I

µ0(α′ + Ωβ′) + γ′mb

]+ TS′

= 0

with ∆∗Υ = ∂2rΥ +

sin θr2

∂θ

(1

sin θ∂θΥ

)

We recover the equation obtained by Soloviev (1967)



Subcase 2 : relativistic Grad-Shafranov equation

Assume dΨ 6= 0 and choose Υ = Ψ (i.e. β = 1).The master transfield equation reduces then to(

1− V−2Wω−Xω2

M2

)∆∗Ψ +

[nhd(hn

)− 1

M2

(dV − 2ωdW − ω2dX

)]· ~∇Ψ

+[ω′

M2 (W +Xω)− C′

C

]dΨ · ~∇Ψ

+µ0σnM2

λ[ΩL′ − E′ + I

µ0(C ′(Ω− ω)− Cω′)

]+ TS′

−λnC (Cξ + Ω Cχ) + I

σM2 [(W +Xω)Cξ + (V −Wω)Cχ] = 0

where C := γ−1, ω := −α, E := Σ/γ and L := Λ/γ

This is the relativistic Grad-Shafranov equation, in the most general form (i.e.including meridional flow and for non-circular spacetimes)



Subcase 2 : relativistic Grad-Shafranov equation

History of the relativistic Grad-Shafranov equation:

Camenzind (1987) : Minkowski spacetime

Lovelace, Mehanian, Mobarry & Sulkanen (1986) : weak gravitational fields

Lovelace & Mobarry (1986) : Schwarzschild spacetime

Nitta, Takahashi & Tomimatsu (1991) : Kerr spacetime (pressureless matter)

Beskin & Pariev (1993) : Kerr spacetime

Ioka & Sasaki (2003) : non-circular spacetimes

Remark: Grad-Shafranov equation not fully expressed by Ioka & Sasaki (2003) +use of additional structure ((2+1)+1 formalism)



Subcase 3 : pure hydrodynamical flow

No electromagnetic field

dΦ = 0 (⇐⇒ α = 0), dΨ = 0 (⇐⇒ β = 0) and I = 0

=⇒ Σ = γ E with E := λh(V −WΩ) and Λ = γ L with L := λh(W +XΩ)

Master transfield + poloidal wind equations:

γ2∆∗Υ + γ2n

hd(h

n

)· ~∇Υ + γγ′dΥ · ~∇Υ

+σn2

h[λ(ΩL′ − E′) + TS′]− γλn (Cξ + ΩCχ) = 0

γ2h2

n2dΥ · ~∇Υ + σh2 −XE2 − 2WEL+ V L2 = 0



Subcase 3 : pure hydrodynamical flowCase of purely rotational motion : γ = 0

The master transfield equation reduces to

ΩL′ − E′ + T

λS′ = 0

A wide class of solutions is found by assuming

Ω = Ω(Υ) andT

λ= T (Υ) with T ′ = −T λL

hΩ′

For Ω = const, this leads to the well-known first integral of motion [T = 0: Boyer

(1965)]

µ

λ=h− TS

λ= const

For Ω 6= const, we obtain instead

ln(µλ

)+∫ Ω

0

F(Ω) dΩ = const

with F(Ω) =W +XΩ

V − 2WΩ−XΩ2(relativistic Poincare-Wavre theorem)



Subcase 3 : pure hydrodynamical flowCase of flow with meridional component : γ 6= 0

Then df 6= 0 and a natural choice for Υ is Υ = f

The master transfield + poloidal wind equations reduces to

∆∗f +n

hd(h

n

)· ~∇f +

σn2

h[λ(ΩL′ − E′) + TS′]− λn (Cξ + ΩCχ) = 0

(22)

h2

n2df · ~∇f + σh2 −XE2 − 2WEL+ V L2 = 0 (23)

with Ω =V L−WE

XE +WL

Given the three functions E(f), L(f) and S(f) and the EOSh = h(S, n), T = T (S, n), (22)-(23) forms a system of coupled PDE for (f, n)

At the Newtonian limit, (22) is the Stokes equation


Conclusion

Outline

1 Introduction





6 Conclusion


Conclusion

Conclusions

Ideal GRMHD is well amenable to a treatment based on exterior calculus.

This simplifies calculations with respect to the traditional tensor calculus,notably via the massive use of Cartan’s identity.

For stationary and axisymmetric GRMHD, we have developed a systematictreatment based on such an approach. This provides some insight onpreviously introduced quantities and leads to the formulation of very generallaws, recovering previous ones as subcases and obtaining new ones in somespecific limits.


Conclusion

Bibliography

J. D. Bekenstein & E. Oron : New conservation laws in general-relativisticmagnetohydrodynamics, Phys. Rev. D 18, 1809 (1978)

V. S. Beskin : MHD Flows in Compact Astrophysical Objects Springer(Berlin) (2010)

B. Carter : Perfect fluid and magnetic field conservations laws in the theoryof black hole accretion rings, in Active Galactic Nuclei, Eds. C. Hazard &S. Mitton, Cambridge University Press (Cambridge), p. 273 (1979)

E. Gourgoulhon : An introduction to the theory of rotating relativistic stars,arXiv:1003.5015 (2010)

E. Gourgoulhon, C. Markakis, K. Uryu & Y. Eriguchi :Magnetohydrodynamics in stationary and axisymmetric spacetimes: a fullycovariant approach, preprint arXiv:1101.3497

K. Ioka & M. Sasaki : Grad-Shafranov equation in noncircular stationaryaxisymmetric spacetimes, Phys. Rev. D 67, 12026 (2003)

S. Nitta, M. Takahashi & A. Tomimatsu : Effects of magnetohydrodynamicsaccretion flows on global structure of a Kerr black-hole magnetosphere, Phys.Rev. D 44, 2295 (1991)


http://arxiv.org/abs/1003.5015

http://arxiv.org/abs/1101.3497

magnetohydrodynamics in stationary and axisymmetric...

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